solve complex equation for $x$ and $y$

Question

How to solve this?

I have tried to put all $z$s on one side,

but I don't have an idea to continue.

$$z^3-i(z-2i)^3=0$$

Answer 1

$$ z^3 = i\cdot(z-2i)^3. $$ Therefore $z$ is one of the three cube roots of $i(z-2i)^3$.

Observe that $(-i)^3 = i$ so $z$ is one of the three cube roots of $(-i)^3 (z - 2i)^3 = (-2-iz)^3$.

One of those three cube roots is $-2-iz$.

So either $z=-2-iz$ or $z$ is one of the other two cube roots.

If $z=-2-iz$ then $z(1+i) = -2$ so $z=-2/(1+i) = i-1$.

Since $z$ is one of the three cube roots of its cube and it's one of the three cube roots of $-2-iz$, the three cube roots of $i-1$ are what we seek.

What are the three cube roots of $i-1 = \sqrt{2}\left(\cos(135^\circ) + i\sin(135^\circ)\right)$. Here we invoke de Moivre's theorem and seek the three numbers that are $135^\circ/3$ and this division is done modulo $360^\circ$. That gives us $$ 45^\circ \text{ or } {45}^\circ \pm \frac{360^\circ} 3 = 45^\circ \text{ or } {-75}^\circ \text{ or } {+165}^\circ. $$ Modulo $360^\circ$, that is the same as $$ 45^\circ \text{ or } 165^\circ \text{ or } 285^\circ, $$ all differing from each other by a third of a circle. So we have: \begin{align} \sqrt 2\left(\cos45^\circ + i \sin45^\circ\right) & = 1+i \\[10pt] \sqrt2 \left(\cos165^\circ + i\sin165^\circ\right) & = \cdots \\[10pt] \sqrt 2\left(\cos 185^\circ + i \sin185^\circ\right) & = \cdots \end{align}

Answer 2

Expand and then find the roots of the polynomial, just like you would for a real polynomial.

Answer 3

Hint: $-i(z-2i)^3 = (i(z-2i))^3$, which is a perfect cube, leaving the left-hand side to be a sum of cubes.

Answer 4

$$z^3-i(z-2i)^3=0\Longleftrightarrow$$ $$(1-i)z^3-6z^2+12iz+8=0\Longleftrightarrow$$ $$(1+i)(z+(1-i))(-iz^2-(2-4i)z+4)=0\Longleftrightarrow$$ $$(z+(1-i))(-iz^2-(2-4i)z+4)=0\Longleftrightarrow$$ $$z+(1-i)=0\Longleftrightarrow \space\space\vee\space\space -iz^2-(2-4i)z+4)=0\Longleftrightarrow$$ $$z=-1+i\Longleftrightarrow \space\space\vee\space\space z^2-(4+2i)z+4i=0\Longleftrightarrow$$ $$z=-1+i\Longleftrightarrow \space\space\vee\space\space z^2-(4+2i)z=-4i\Longleftrightarrow$$ $$z=-1+i\Longleftrightarrow \space\space\vee\space\space z^2-(4+2i)z+(3+4i)=3\Longleftrightarrow$$ $$z=-1+i\Longleftrightarrow \space\space\vee\space\space (z-(2+i))^2=3\Longleftrightarrow$$ $$z=-1+i\Longleftrightarrow \space\space\vee\space\space z-(2+i)=\pm\sqrt{3}\Longleftrightarrow$$ $$z=-1+i \space\space\vee\space\space z=(2+i)\pm\sqrt{3}$$